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Question #40828

Difficulty: 8/10 | 🟢 Free

The Question

A mirror surface $S$ is parameterized by $\vec{r}(u,v) = u^3 \hat{i} + v^3 \hat{j} + (2u^3 - 3v^3 + u^3 v^3) \hat{k}$ for $u, v \in \mathbb{R}$ . A light ray with incident direction vector $\vec{v}_i = \hat{i} + 2\hat{j} + 3\hat{k}$ strikes $S$ at the origin. Find the reflected direction vector $\vec{v}_r$ such that $|\vec{v}_r| = |\vec{v}_i|$ .

Options

A.
$3\hat{i} - \hat{j} + 2\hat{k}$
B.
$-\hat{i} + 5\hat{j} + 4\hat{k}$
C.
$\hat{i} + 2\hat{j} - 3\hat{k}$
D.
$\vec{0}$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Bridge

Practice problems that bridge the gap between foundational theory and advanced practical application.

Concepts Behind This Question

Reflective Property|Dot Product Projection Sign|Parametric form pitfalls

Why This Question Is Tricky

Common Mistakes

This question tests a student's ability to navigate parametric singularities and apply vector reflection formulas rigorously.

The primary trap lies in the surface parameterization: computing the normal vector via the standard cross product of partial derivatives ( $\vec{r}_u \times \vec{r}_v$ ) yields the zero vector at the origin.

Students who blindly apply this formula may incorrectly conclude the normal is undefined or make algebraic errors attempting to use limits.

The intended, elegant path requires recognizing the implicit Cartesian equation of the surface to find the non-zero gradient vector.

Furthermore, the problem tests the correct application of the dot product projection in the reflection formula, ensuring students handle the negative sign of $\vec{v}_i \cdot \vec{n}$ correctly without being confused by the normal's orientation.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Conic Sections - Class 11
Concept	Reflective Property
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free